Geodesics on homogeneous spaces is a classical topic in Riemannian geometry with several recent contributions by many authors, see for instance[1]}, [2]}, [3]}, [4]} and references therein. In this paper we will consider Riemannian metrics preserving some symmetries of the homogeneous space, but not all transitive group \(G\) . More specifically, we will consider Riemannian metrics invariant by the maximal compact subgroup \(K\subset G\) . In this context we cannot apply directly the standard Lie Theory techniques for the study of geodesics since the Riemannian metrics are not invariant for the whole group \(G\) . Our strategy is the following: we start with the sequence of identifications
\(\operatorname{\mathrm {Ad}}(G)H_0 \overset{{\cong }}{\longrightarrow } T^\ast \operatorname{\mathrm {Ad}}(K)H_0 \overset{{\cong }}{\longrightarrow } T \operatorname{\mathrm {Ad}}(K)H_0,\) 
